Note that the original TV term is kept and hence the objective functional in (REF ) is nonsmooth. Obviously, (REF ) has no closed-form solution and it should be solved iteratively by a certain algorithm. Also, the dimension of \(p_h\)  is the same as that of \(q_h\) , and it may be high for a higher-dimensional space and fine mesh discretization. For instance, it is of order \(10^6\)  if the mesh size \(h=1/1024\)  for the unit square \(\Omega \subset \mathbb {R}^2\) . Hence, it is also necessary to consider how to solve the \(p_h\) -subproblem () efficiently for implementing the ADMM (REF ). We reiterate that it is always more preferable to choose some model-tailored algorithms in accordance with the structure of the problem under consideration. For the \(p_h\) -subproblem (REF ), certainly it can be treated as a generic optimization problem and then some generic-purpose or less structure-exploiting algorithms can be applied. But it turns out that the deep convolutional neural network (CNN), which has been significantly enhanced in recent literatures (e.g., [1]}, [2]}), is a much better choice for the \(p_h\) -subproblem (REF ). Below is the detail.
